True or False ( Countability)

Question

Let $A$ and $B$ be non empty, define $B^A = \{F|F : A \rightarrow B\}$, and $A^B = \{F|F : B \rightarrow A\}$. (Basically $A^B$ and $B^A$ are the number of functions for $F:A \rightarrow B$ and $F:B \rightarrow A$ respectively.)

Then is the following:

$B^A\,\&\, A^B$ are both countable $\implies$ $A\, \& \,B$ are both countable.

true or false.

I know that if $A\, \& \,B$ are countable then $A^B$ and $B^A$ need not be countable. But the given statement is the converse. How to determine if it is true or false?

Answer 1

Hint: If $A$ is an arbitrary set and $B$ is non-empty then there is an injection $A \to A^B$.

Answer 2

For each element $b\in B$ define a constant function from $A$ to $B$ defined by f(x)=b.

The cardinality of this set of functions is simply cardinality of B.

Thus B is countable because the above mentioned set is a subset of $B^A$.

Similarly we can prove that A is countable.